DEFINITION TRANSVERSE ELECTRIC (TE) WAVES OR H WAVES IN PARALLEL PLANES:

In the case, the component of electric field vector E lies in the plane transverse to the direction of propagation that is there is no component of E along the direction of propagation where as a component of magnetic field vector H lies along the direction of propagation.

Derivation of transverse electric waves in parallel planes:

As the direction of propagation is assumed as z-direction, therefore

E_z = 0 and H_z is not equal to 0

Now by substituting E_z = 0 in equation (8) of article “waves between parallel planes”, we get

E_x= 0 and H_y = 0 and

E_{y not equals to} 0 , H_x  not equals to 0

Now write wave equations for free space in terms of E

ѲE =g²_gE

= -w²μεE                     (because g²_g = (jwμ) (σ + jwε) As σ =0(from                                                                            assumption  (c) of article “waves between parallel planes” => g²_g =-w²με)

Or                    d²E/dx² + d²E/dy² + d²E/dz² = -w²μεE

For the y component, the wave equation will become

d²E_y/dx² + d²E_y/dg ² + d²E_y/dz² = -w²μεE_y

also                       dE_g/d_g  = 0                                                      [using assumption (e)]

and                       E_g = E_g0 e^– g_g^z                                                                    [using assumption (f)] of article “waves between parallel planes”

Thus           d²E_g/dz² = g²Eg²_gEg²_gE_g

By substituting above values of dE_g/d_g and d²E_y/dz² in wave equation, we get

d²E_g/dx²  + g²_gE_g= -w²μεE_g

or              d²E_g/dx²  = -(g²_g + w²με) E_g

or              d²E_g/dx²  = –K²_g E_g                                                                  ..(9)

where      K²_g = g²_g + w²με

as              E_g = E_g0  e^– g_g^z                                                                          ..(10(a))

thus           d²E_g/dx²  = d²E_g0 e^– g_g^z/dx²                                                                    ..(10(b))

by substituting equation(10) in equation (9) , wave equation becomes

d²E_g0/ dx² = K²_g E_g0 

The above equation is a standard differential equation of simple harmonic motion and its solution can be written in the form

E_g0 = A₁ sin K_gx + A₂cos K_gx

Or              E_g = (A₁ sin K_gx + A₂ cos K_gx) e^– g_g^z                                      ..(11)

Where A₁ and A₂ are arbitrary constants.

A₁ and A₂ can be determined with the help of following boundary conditions :

E_tan = 0 at the surface of conductor

This implies that

                              E_y = 0 at x = 0

E_y = 0 at x = a

Thus apply boundary condition

That           E_y = 0 at x = 0

So equation (11)will become

0 = (A₁ sin K_g0 + A₂ cos K_g0) e^– g_g^z

Thus equation (11) reduces to

E_y = A₁ sin K_gxe^– g_g^z                                                   ..(12)

Now apply boundary condition

That E_y = 0 at x = a in equation, we get

0 = A₁ sin K_gae^– g_g^z

Or              sinK_ga = 0 as A₁¹ 0

Or              K_ga = mπ

Or              K_g    = mπ/a, wher m = 1,2,3

(If m = 0, all field components vanish. Its a special case and will be discussed later)

.           equation (12)becomes

. .         E_y = A₁ sin (mπ/a)e^– g_g^z

E_y = A₁ sin (mπ/a)e^–jb^gz       (. . g_g =  jb_g  if α_g = 0)             ..(13(a))

Put equation  (13 a)in (8 d of article “waves between parallel planes”)and integrate

A₁ sin (mπx/a)e^– g_g^z = jwμ/ K²_g dH_z/dx   dx

H_z = A₁ K²_g e^– g_g^z /jwμ          sin(mπx/a)/dx

= – A1m²π²/a2   e^– g_g^z/ jwμ  cos (mπx/a)/ mπ      where K²_g = m²π²/a²

H_z =- A₁ mπ cos (mπx/a) e^– g^gz/ jwμ

or                H_z =- A₁ mπ cos (mπx/a) e^– g^gz/ jwμ                       (g_g = jb_g)          ..(13(b))

Differentiate (13(b))w.r.t.  x

dH_z/dx  = A1m²π² sin (mπx/a) e^– g^gz/ jwμa²

Substitute above equation in equation

H_x=- g_g / K²_g A1 m²π²/ jwμa² sin (mπx/a) e^– g_g^z

H_x=- g_g m²π²/a²         A1 m2π2/jwμa2     sin (mπx/a) e^– g_g^z

H_x=- g_g  A₁ /jwμ   sin (mπx/a) e^– g_g^z

H_x=- b_g  A₁ /wμ   sin (mπx/a) e^– g_g^z                 ( g_g =  jb_g)        ..(13(c))

Equation (13(a)), (b), (c)) represent the expressions for field components for TE waves. Each value of m in equations (13) represent a particular field configuration and the wave associated with integer m is designated as TE_m wave. The lowest order mode that can exist in this case is TE₁ mode.

This is the definition, discussion and derivation of transverse electric (TE) waves between parallel planes or plates.